Difference between revisions of "1952 AHSME Problems/Problem 23"
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| + | == Problem== | ||
| + | If <math> \frac{x^2-bx}{ax-c}=\frac{m-1}{m+1} </math> has roots which are numerically equal but of opposite signs, the value of <math> m </math> must be: | ||
| + | |||
| + | <math> \textbf{(A) \ }\frac{a-b}{a+b} \qquad \textbf{(B) \ }\frac{a+b}{a-b} \qquad \textbf{(C) \ }c \qquad \textbf{(D) \ }\frac{1}{c} \qquad \textbf{(E) \ }1 </math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | Cross-multiplying, we find that <math> (m+1)x^2-(bm+am+b-a)x+c(m-1)=0 </math>. Because the roots of this quadratic are <math> r_1 </math> and <math> -r_1 </math>, their sum is <math> 0 </math>. According to [[Vieta's Formulas]], <math> \frac{bm+am+b-a}{m+1}=0 </math>, or <math> m(a+b)=a-b </math>. Hence, <math> m=\boxed{\textbf{(A)}\ \frac{a-b}{a+b}} </math>. | ||
| + | |||
| + | ==See also== | ||
| + | {{AHSME 50p box|year=1952|num-b=22|num-a=24}} | ||
| + | {{MAA Notice}} | ||
Revision as of 22:53, 24 January 2014
Problem
If 
has roots which are numerically equal but of opposite signs, the value of 
must be:
Solution
Cross-multiplying, we find that 
. Because the roots of this quadratic are 
and 
, their sum is 
. According to Vieta's Formulas, 
, or 
. Hence, 
.
See also
| 1952 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
